Maxine Peake slams and slides as the shape-shifting Skriker. Photograph: Christopher Thomond for the Observer

She would not be welcome as a member of the Garrick Club. She has no penis, no establishment position and is not big on banter. Nevertheless, the Skriker is one of the primary figures of modern theatre.

As Caryl Churchill’s shape-shifting, doom-wreaking fairy, Maxine Peake rams home this importance. She slams and slides and swarms. She comes on as a crop-haired, grey-clad prophetess, growling accusations. She reappears, whining, as a shaggy creature tethered to plastic bags. She becomes a sleek woman from a southern state, with shades and a cocktail glass, and a clingy, wheedling girl in an anorak. She is a tattered, winking Gloriana, a sleek, androgynous seducer in a tie, and a winsome elf with a teeny voice and gauzy wings.

Churchill’s play is almost entirely female. The voice of its ancient Cassandra is dominant. Its most sympathetic characters are two young women, strongly rendered by Laura Elsworthy and Juma Sharkah. One has killed her baby; the other is pregnant. The Skriker haunts them, tormenting and enticing. The few males in Sarah Frankcom’s explosive production are part of a disordered landscape in which animation means mutation: one who writhes in ecstatic dance may be partly a horse; another has a giant ear sprouting from the top of his head like a satellite dish.

Yet The Skrikerreclaims what have been thought as “women’s issues” for humanity. Motherhood may, after all, also affect men. Churchill uses a female voice to express a skewed world. And what better time to stage this? We are in an era of theatrical dystopias. Of dark fragmentary dramas, which dip in and out of underworlds. A few months ago, Simon Stephens’s Carmen Disruption smouldered at the Almeida. Alistair McDowall’s tale of lost souls, Pomona, will shortly be seen at the National and in Manchester, on whose geography it draws. Zinnie Harris’s How to Hold Your Breath, which also starred Peake, made a claim on the same territory at the Royal Court. The Skriker, first staged in 1994, now looks like the fairy godmother of them all.

It is powerful in picturing disorder. It is bewildering, sometimes maddening in its fecund confusion. It is also extraordinarily prescient. Using fairytale to project hard truths is now common feminist currency; it was rarer 20 years ago. As was certainty about climate catastrophe, an environmental tragedy that here looks like moral rupture, psychic disaster writ large. Weird things are happening with the weather. “It was always possible to think whatever your personal problem, there’s always nature... Nobody loves me but at least it’s a sunny day.” That consolation has now gone.

Everything is disintegrating, including speech. The Skriker’s language freewheels from sense to delirium. It is as if the speaker had a tempest in her mouth that blows the boundaries between one word and another: “Pin prick cockadoodle do you feel it?” It is not an invented language; rather a repunctuation. Her opening speech, delivered in a sustained rush, is almost the length of a short Beckett play and has some of the same force. Peake can’t make its sentences clear, but she makes it evident that, however hermetic her outpouring, it is not all bunkum. It swims in and out of sense: “The baby has no name better nick a name, better Old Nick than no name.” She pulls you into the echoes of nursery rhyme and fairytale. Specially commissioned music by Nico Muhly and Antony Hegarty (of Antony and the Johnsons), sometimes harsh, sometimes gently marimba, also penetrates these speeches.

Lizzie Clachan, one of the theatre’s boldest design talents, makes a bedlam cabaret out of the Royal Exchange. Audience members sit at rough wooden tables amid the duskily lit action. In a terrific banquet scene, in which huge platters are laden with goodies, one guest delivers a running commentary on what it is to see her own limbs and parts spread out to be devoured. Around the stage, alcoves contain glimpses of ordinary life, diminished to miniature size: rows of sunflowers, bright little houses.

It is extraordinary how rapidly Manchester international festival has established itself. Manchester and Dublin are now the cities for guaranteed festival excitement, not least because the programme is not all one-off fizzing. It looks to the future. The Skriker shows in action one of the most interesting of theatrical partnerships. Frankcom, who runs the Royal Exchange, directed Peake as Hamlet and – incandescently – in her recitation of The Masque of Anarchy. This latest collaboration proves that her theatre will go on provoking.